# What fraction of a given sample of a radioactive nuclide remains after four half-lives?

Jun 25, 2016

It is $\frac{1}{16}$ of the original population.
At the beginning you have that the number of atoms is ${N}_{0}$.
After the first half lives you have ${N}_{1} = {N}_{0} / 2$ because by definition the half life is that time where half of the sample had the decay.
After the second half life you have ${N}_{2} = {N}_{1} / 2 = {N}_{0} / 4$.
As you can see, after each lifetime you divide by two the original population, then after four times you divide four times for $2$ that means that you divide by ${2}^{4} = 16$.
So the final population ${N}_{4} = {N}_{0} / 16$.